Low PAPR Spatial Modulation for SC-FDMA

Transcription

1 Low PAPR Spatial Modulation for SC-FDMA Darko Sinanović, Gordan Šišul, Borivoj Modlic Abstract Although spatial modulation (SM) presents an attractive multiple-input multiple-output (MIMO) technique, it increases peak-to-average power ratio (PAPR) on all transmit antennas when applied directly to single-carrier frequency division multiple access (SC-FDMA). Therefore, we propose an SM modification, named low PAPR SM (LPSM), to preserve the low PAPR level of SC-FDMA and to achieve the benefits of SM. In addition, as optimal maximum likelihood detection (MLD) receiver has very high computational complexity when applied to SC-FDMA system, we observed suboptimal receivers for SC- FDMA with LPSM. Besides low complexity linear receivers, we proposed and analyzed two receivers: near-mld and improved minimum mean-square error (immse) receiver. Near-MLD receiver achieves performance very near to MLD with significantly lower complexity, whereas immse presents a tradeoff between MLD and linear receivers in terms of performance and complexity. Index Terms spatial modulation, SC-FDMA, MIMO, PAPR, MLD. S I. INTRODUCTION INGLE carrier frequency division multiple access (SC- FDMA) is well known as a modulation and a multiple access technique, selected for the uplink of Long Term Evolution (LTE) mobile communications standard [1]. It has been shown that it offers similar performance to widely adopted orthogonal frequency division multiplex (OFDM), while maintaining much lower peak-to-average power ratio (PAPR) [, 3]. PAPR is a measure of fluctuations in the output power and presents an important property of a modulation technique from the implementation point of view. High PAPR places constraints on the output power as it is necessary for the amplifier to retain in the linear operation range in order to avoid distortions of the transmitted signal [4]. This requires the amplifier operation below saturation, reducing its efficiency. If it is not the case, the clipping occurs and it generates unwanted in-band distortion and out-of-band radiation. Therefore, to avoid the clipping, the modulations with high PAPR require the amplifier with wide linear range, increasing the cost of the transmitter [4]. SC-FDMA is usually implemented as discrete Fourier transform (DFT) precoded OFDM, thus it is of the similar Submitted on October 1th 015. Copyright (c) 015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. D. S. is with the Independent System Operator, Sarajevo, Bosnia and Herzegovina ( d.sinanovic@nosbih.ba). G. S. and B.M. are with Faculty of Electrical Engineering and Computing, Zagreb, Croatia ( gordan.sisul@fer.hr, borivoj.modlic@fer.hr). complexity. The precoding decreases the PAPR level of the output signal to the level of the single carrier modulations signals, without greater impact on the performance. This significantly relaxes the constraints for the output amplifier, making SC-FDMA an attractive OFDM alternative, especially for mobile communications uplink. As it is based on OFDM, simple one-tap per subcarrier frequency domain equalization (FDE) used in OFDM can be applied to SC-FDMA as well [5]. SC-FDMA is suitable for even further PAPR level decrease, especially for the high order modulations, using different PAPR reduction techniques [6-8]. All multiple-input multiple-output (MIMO) techniques applicable for OFDM can be applied to SC-FDMA in a similar fashion [9]. However, it has to be taken into consideration the effect of any MIMO operation performed at the transmitter side to the PAPR level of the signals on all transmit antennas. If PAPR levels are increased, it directly affects the transmitter cost and SC-FDMA loses its main advantage over OFDM. Therefore, MIMO techniques for SC-FDMA have to maintain PAPR level at the acceptable level. Spatial modulation (SM) has recently gained attention as a low complexity MIMO technique. It conveys information using the index of the transmit antenna and the quadrature amplitude modulation (QAM) or phase shift keying (PSK) symbol. Instead of increasing the constellation size, it increases the bitrate using the index of the transmit antenna to carry the additional information. Initially, it was proposed for single-carrier QAM/PSK modulation in flat fading channels [10]. In essence, one part of the bits is used for QAM/PSK modulation and the other part of bits is used to select which antenna is used for the transmission of the corresponding QAM/PSK symbol. When one antenna is selected, all other antennas are inactive, i.e. they transmit zeros. In that case, the transmitter has one radio-frequency (RF) chain and the antenna switching is performed after the amplifier, before the signal is applied to the selected antenna. The transmitter complexity is very low and the performances in Rayleigh channel models are even better than those of the transmit diversity or spatial multiplexing techniques, such as spacetime block coding (STBC) or V-BLAST, for the same bitrate [10]. Later, it has been shown that significant improvement is achieved when the optimal joint maximum likelihood detection (MLD) receiver is used (it makes the decision jointly for QAM/PSK symbol and the antenna index) [11]. Further, it has been shown in [1] that due to the pulse shaping, a necessary part in practical implementation, the number of RF chains has to be greater than one, but still less than the number of the transmit antennas. Detailed overviews of SM are given in [13, 14].

2 Using the similar approach, space shift keying (SSK) uses only the index of the transmit antenna for the transmission of the information [15]. In that case, no underlying modulation is used. As the narrow subchannels in OFDM can be observed as flat fading channels, the migration to OFDM is straightforward [16]. Again, it has been shown that for different underlying modulations, bitrates and antenna configurations, SM generally outperforms other MIMO techniques for the same bitrate. Besides good performances, SM in OFDM shows greater transmitter complexity, compared to [10], as the transmitter is required to have as many RF chains as the transmit antennas [17] and the advantage of the single RF chain transmitter is lost. The antenna switching is not performed in the time domain (TD), but in the frequency domain (FD), prior to the inverse fast Fourier transform (IFFT). PAPR of SM in OFDM has been analyzed recently in [18]. Single carrier modulations with cyclic prefix [17, 19] or zero padding [0], added to combat frequency selective fading, with SM have been observed recently. Unlike OFDM, it has been shown that the simple transmitter with single RF chain can be used and the performances in frequency selective channels are acceptable. Again, due to the pulse shaping [1], the number of RF chains in practical implementations is greater than one. Later, generalized SM (GSM) implemented in these single carrier modulations has been observed in [1, ] and multiuser MIMO for SM with cyclic prefix was analyzed in [3]. Very recently, a detailed overview of SM in single carrier modulations has been given in [4], addressing the issue of PAPR level and the pulse shaping. An SM alternative, named space-time shift keying (STSK), that uses more than one active antenna by selecting one of the predefined dispersion matrices, applied to SC-FDMA has been studied in [5, 6]. Due to inherent complexity of the optimal MLD receiver in SC-FDMA, the low complexity alternatives for the receiver have been proposed. It has been shown that even with simple linear equalization, such as minimum meansquare error (MMSE) equalization, SM in SC-FDMA presents an attractive MIMO technique. However, for the best of our knowledge, the PAPR for SC- FDMA with SM has not been studied yet. In this paper we explore the PAPR problem of the conventional SM in SC- FDMA and confirm the PAPR increases on all transmit antennas when SM is directly applied to SC-FDMA. To avoid this, and to maintain the advantages of SM, in this paper we propose low PAPR SM (denoted as LPSM) for SC-FDMA. In this paper, two and four transmit antenna cases are observed. Although joint MLD receiver presents the optimal receiver for SM [11], it inherently suffers from very high complexity in SC-FDMA systems because the detection is performed jointly per whole SC-FDMA symbol [7]. Hence, in this paper only short overview of MLD receiver is given and suboptimal receivers are studied and the two novel receivers for LPSM are proposed. In Section II, after the short overview of SC-FDMA and SM, the generalization of SM and the design criteria for LPSM are given. Section III gives an overview of the optimal MLD receiver, linear receivers and the proposal of the two receivers with decreased complexity and the performance near the optimal receiver. The complexities of the observed receivers are discussed in Section IV. The outcomes of the simulations performed for the comparison of the PAPR curves for SM, LPSM and the conventional SC-FDMA and for the performance comparison in different channel models are given in Section V. The paper is concluded in Section VI. The following notation is used in the paper: italic symbols denote scalar values; bold symbols denote vectors/matrices; ( ) T and ( ) H are transpose and Hermitian vector/matrix; F M and I M present M M Fourier and identity matrices respectively; diag( ) is a diagonal matrix of a vector; mod( ) denotes the modulo operation and F presents Frobenius norm of a vector/matrix. II. SM FOR SC-FDMA First of all, SM for two transmit antennas is observed. The transmitter block scheme is presented in Fig. 1. The underlying modulation is QAM (or PSK) with constellation size L and the user occupies M out of N available subcarriers. Input bit sequence, consisting of M(L + 1) bits, is divided into two groups. The first group of ML bits is mapped to M QAM M symbols sequence {s } i=1, and the remaining M bits, {b M SM } i=1, are used for SM. Fig. 1. The block scheme of the SC-FDMA SM two antenna transmitter. SM is performed in the time domain. SM is performed in a way that the each bit in the second sequence selects the antenna that will transmit the corresponding QAM symbol (if b SM = 0, then s is transmitted via antenna 1, otherwise via antenna ). The other antenna is inactive, i.e. it transmits zero instead of the QAM symbol. The two blocks, the each of length M, consisting of QAM symbols and zeros, are converted to the FD using M- point DFT operations. In LTE, SC-FDMA is implemented as localized SC-FDMA [1] that assigns the block of M subsequent subcarriers, out of N possible, to one user. The transmitter sets other N M inactive subcarriers to zero. Therefore, the two blocks of length M in the FD are zero-padded to obtain two blocks of length N. It is worth noting that M is the number of the subcarriers assigned to the user and, equivalently, the number of QAM/PSK symbols inside one SC-FDMA symbol. After the zero-padding, N-point inverse fast Fourier transform (IFFT) blocks generate the signals in the TD and prior to the transmission, cyclic prefix (CP) is added. CP length has to be greater than the channel delay spread to avoid intersymbol interference between the consecutive SC-FDMA symbols. The

3 blocks for SM are inserted in the TD after QAM/PSK modulation and prior to M-point DFT [0]. Analogously to OFDM, SM can be implemented in the FD, between M-point DFT and N-point IFFT. In that case, SM is performed over the samples in the FD, which are obtained after M-point DFT performed over M QAM/PSK symbols. Hence, the each sample in the FD is transmitted over the active antenna, while zero is transmitted over the other. As some of these samples can be equal to or near to zero, the receiver cannot determine which antenna is the active as the both antennas transmitted zeros on the observed subcarrier. Hence, SM in the FD inevitably suffers from the residual bit error rate (BER) on SM bits. Therefore, SM in the TD, as presented in Fig. 1., is observed in the paper. At the receiver, which can have more than one receive antenna, the signals undergo symmetric process. Its block scheme is not presented as it depends on the type of the detection and/or equalization. It is further discussed in Section III. The four transmit antenna case is very similar. Instead of observing one SM bit for antenna selection, the transmitter uses two SM bits to select the one of the four available transmit antennas. Other three antennas transmit zeros. SM is initially proposed for a single carrier modulation in a flat fading channel. If the transmitter has N T transmit antennas, every QAM/PSK symbol is transmitted over exactly one of them [10]. During that symbol interval, all the other antennas are inactive. In that case, the transmitter uses only one RF chain and SM switching is performed in the TD, after the amplifier, therefore the PAPR level at the amplifier is the same as without SM. Usually, N T is selected as T, where T is the number of SM bits conveyed via antenna index. As shown in Fig. 1., SC-FDMA SM transmitter performs SM prior to the amplifying, therefore the transmitter has N T RF chains, so the PAPR levels for all N T transmit antennas have to be observed. This is due to the fact that although SC- FDMA is essentially a single carrier modulation, it is implemented as DFT precoded OFDM, thus it requires multiple RF chains for SM, as it is the case in OFDM SM [16]. Therefore, the advantage of single RF chain in SM is lost in OFDM and SC-FDMA. As DFT and IDFT are dual operations and N is greater than M, SC-FDMA SM signals can be observed as an oversampled single-carrier SM QAM/PSK signals. The insertion of zeros in the TD, instead of QAM/PSK symbols, decreases the average power on all transmit antennas. This results in increase of PAPR. If the transmitter has two transmit antennas, it can be expected that the half of the QAM symbols for one antenna will be replaced by zeros and the average power level will be decreased by a half. The exact PAPR level is obtained via simulations in Section V. A. LPSM for two transmit antenna case In order to avoid the increase of PAPR, we propose the modification of SM, named LPSM (Low PAPR SM). First, we used the generalization of SM. As the transmitter uses N T RF chains, it is not necessary that the other antennas are inactive, while the selected one is active. The two transmit antenna case is observed first. Let a, b, c, and d present the complex coefficients that will be determined in order to preserve the same PAPR level on all antennas. The generalized SM coding scheme is given in Table I. For conventional SM, a = d = 1 and b = c = 0. Table I. Generalized SM coding scheme, two transmit antennas Antenna 1 Antenna b SM = 0 as bs b SM = 1 cs ds These coefficients can be obtained by setting three conditions. The first condition is to preserve the same total output power level as in the conventional SM, (1) and (). For convenience, unitary total power level is assumed. The second condition, (3) and (4), is that a change in an SM bit does not generate the power fluctuation. This condition is low PAPR condition. Finally, as SM is assumed to be an open-loop MIMO technique, the equal average power allocation on all transmit antennas is the third condition (5). a + b = 1 (1) c + d = 1 () a = c (3) b = d (4) 0.5( a + c ) = 0.5( b + d ) (5) It can be easily shown that: a = b = c = d = 0.5 (6) From (6), it can be concluded that all four coefficients have the same amplitude ( 0.5), whereas the phases have to be found. As the transmitter does not have channel state information, the difference between the phases of the coefficients for antenna 1 and antenna is irrelevant. Actually, only the phase change between a and c (antenna 1) and b and d (antenna ) is relevant. Without loss of generality, it can be assumed that the phases of a and b are 0. Thereby, we can write: a = b = 0.5 (7) c = 0.5e jα (8) d = 0.5e jβ (9) where α and β are two real phases. To select the coefficients, the optimum values for α and β have to be found. As the channel state information is not known by the transmitter, the optimum values are obtained from the condition that the minimum distance between any two possible transmitted pairs is maximized. If s is an QAM/PSK symbol from the defined constellation with L symbols, the transmitter conveys L pairs [as, bs] or L pairs [cs, ds]. Hence, the minimum value of the distances between any two of L+1 pairs has to be maximized. The two cases can be distinguished: 1) the distance between the two different pairs when SM bits are the same. Let s i and s j are two different QAM/PSK symbols. The squared distance between any two pairs is: D = as i as j + bs i bs j = ( a + b ) s i s j = s i s j (10) if both SM bits are zero. Otherwise,

4 D = cs i cs j + ds i ds j = ( c + d ) s i s j = s i s j (11) Obviously, the distance is the same as between the corresponding QAM/PSK symbols regardless of α and β. ) the distance between the two pairs when SM bits are different. Let s i and s j are two, not necessarily different, QAM/PSK symbols. The squared distance between any two pairs is: D = as i cs j + bs i ds j = 0.5 ( s i e jα s j + s i e jβ s j ) (1) We are observing the sum of the squared distances between s i and e jα s j and between s i and e jβ s j. Let us first observe the case when both symbols s i and s j are of the same absolute value. These three complex values can be observed as three points on a circle of diameter s i = s j. As QAM/PSK constellations are symmetric to the origin, there is also a symbol s i = e jπ s i. Therefore, there are four points on the circle, as seen in Fig.. Fig.. Symbols s i, e jα s j, e jβ s j and s i in complex plane. Using the law of cosines, (1) can be written as: D = 0.5 e jα s j e jβ s j + s i e jα s j s i e jβ α β s j cos ( ) (13) D = 0.5 s j e jα e jβ α β + Pcos ( ) (14) D = s j α β (1 cos (α β)) + Pcos ( ) (15) where P = s i e jα s j s i e jβ s j is non-negative for any α and β. The squared distance for the symmetric symbol from e jα s j and e jβ s j is: D symm = 0.5 ( s i e jα s j + s i e jβ s j ) (16) D symm = 0.5 e jα s j e jβ s j + s i e jα s j s i e jβ s j cos (π α β ) (17) D symm = s j α β (1 cos (α β)) Qcos ( ) (18) where Q = s i e jα s j s i e jβ s j is non-negative for any α and β. From (15) and (18), it can be seen that the first part is the same and it has the maximum value when: cos(α β) = 1, that is when α = β + (K + 1)π, where K is an integer. For the second part of (15) and (18), it can be noted that Pcos ( α β ) and Qcos (α β) cannot be both positive for the same α and β. Thus, the maximum of the minimum of the both second parts is obtained when cos ( α β ) = 0, implying α = β + (K + 1)π, which is the same as for the maximum of the first part. Finally, it can be concluded that the maximum value for the minimum distances are obtained when α = β + (K + 1)π. This implies that the actual phases are not relevant, but rather the difference between phases for c and d. Finally, without the loss of generality, it can be written a = b = 0.5 (19) c = 0.5e jα (0) d = 0.5e j(α+π) = c (1) The phase α is irrelevant for the maximum of the minimal distance, so it can be selected α = 0, but in that case only antenna transmits the information on SM bits. If the antenna has severe channel conditions, SM bits will have significantly worse performance than QAM/PSK bits. Therefore, we recommend the selection of the parameter α depending on the underlying modulation in a way that the information on SM bits is transmitted via both antennas. It can be defined as the half of the smallest phase shift between two different symbols of the same absolute value. For example, if the modulation is QPSK, then the parameter can be selected as α = π to introduce the phase shift on the both antennas. 4 In (1), if the symbols s i and s j have different absolute values, it is obvious that the distance is greater than in the case when they have the same absolute value, so it is not the minimum distance. From (15) and (18), using the conclusion that α = β + (K + 1)π, it can be noted that the minimum distance between two pairs if SM bit is different and if both QAM/PSK have the same absolute value is D 1 = 0.5 s j. On the other hand, if SM bit is the same, using (10) or (11), the distance is D = s i s j. Obviously, these distances depend on the underlying modulation, but, if conventional QAM or PSK modulation is used it can be concluded that D 1 < D. E.g. for QPSK is D 1 = 0.5 s j and D = s j. This implies that the distance between the two pairs when SM bit is different is significantly smaller. Hence, it can be expected that when one pair is received with an error, that the SM bit is more likely with error. This conclusion is used for the proposed receivers in Section III. Finally, the LPSM coding scheme can be defined as in Table II., where α is selected as the half of the smallest phase shift between two different symbols of the same absolute value in the underlying constellation. This coding scheme is implemented at the transmitter in the blocks presented in Fig. 1. as SM for antennas 1 and.

5 Table II. LPSM coding scheme, two transmit antennas Antenna 1 Antenna b SM = 0 0.5s 0.5s b SM = 1 0.5e jα s 0.5e jα s B. LPSM for four transmit antenna case For the case of the four transmit antennas, the similar approach can be used. The generalized scheme, similar to Table I, is given in Table III. For the conventional SM, a 1 = b = c 3 = d 4 = 1, and all other coefficients are 0. Table III. Generalized SM coding scheme, four transmit antennas Ant.1 Ant. Ant.3 Ant.4 ) = (0,0) a 1 s b 1 s c 1 s d 1 s ) = (0,1) a s b s c s d s ) = (1,0) a 3 s b 3 s c 3 s d 3 s ) = (1,1) a 4 s b 4 s c 4 s d 4 s Table IV-1. LPSM coding scheme, four transmit antennas, antennas 1. and. Ant.1 Ant. ) = (0,0) 0.5s 0.5s ) = (0,1) 0.5e jα s 0.5e j(π +α) s ) = (1,0) 0.5e jα s 0.5e j(π+α) s ) = (1,1) 0.5e j3α s 0.5e j(3π +3α) s Table IV-. LPSM coding scheme, four transmit antennas, antennas 3. and 4. Ant.3 Ant.4 ) = (0,0) 0.5s 0.5s ) = (0,1) 0.5e j(π+α) s 0.5e j(3π +α) s ) = (1,0) 0.5e jα s 0.5e j(π+α) s ) = (1,1) 0.5e j(π+3α) s 0.5e j(π +3α) s Using the similar approach as in the two transmit antenna case, it can be concluded that a k = b k = c k = d k = 0.5 () for k = 1,,3,4. For the two transmit antenna case, if we assume that α = 0, the phase shift between two SM bits for the antenna 1 is 0 and for the antenna is π. In analogy to two transmit antenna case, it can be generalized that for antenna t, which is one of N T antennas, the phase shift φ t between consecutive SM possibilities can be presented as: π(t 1) (3) φ t = N T Using (3) for four transmit antenna case, the optimum values are obtained when: a k = 0.5 (4) (k 1) (5) b k = 0.5e jπ c k = 0.5e jπ (k 1) (6) d k = 0.5e j3π (k 1) (7) Again, the phase shift for the each SM bit pair can be introduced in order to avoid the fact that the information of SM bits is not transmitted over antenna 1 and that antenna 3 does not transmit the information on b SM,1. Hence, the final LPSM coding scheme is given in Table IV. In analogy to the two transmit antenna case, α is selected as the quarter of the smallest phase shift between two different symbols of the same absolute value in the underlying constellation. A. Optimal MLD receiver III. LPSM RECEIVER If the receiver has N R receive antennas, let Y present an N R 1 vector of the N R received signals on all antennas in the FD on the i th subcarrier and let H present an N R N T channel matrix for the observed i th subcarrier. Let U present an N T M matrix of the transmitted samples in the FD of one SC-FDMA LPSM signal over all transmit antennas. The set of all possible transmitted signals, i.e. matrices U, is denoted as A. If U presents an i th column vector of U, consisting of N T samples in the FD transmitted on i th subcarrier, using the MLD approach, the matrix U that minimizes the distance: M argmin Y H U (30) F U A i=1 is selected by the receiver. The main problem is that one SC-FDMA LPSM symbol conveys M QAM/PSK symbols, with each carrying L bits, and MT SM bits, hence the set A has M(L+T) possible SC-FDMA LPSM symbols. This requires that (30) has to be calculated M(L+T) times per SC-FDMA LPSM symbol, making practical usage of the MLD receiver impossible. B. Linear equalization receiver The receivers with linear equalization (LE), zero-forcing (ZF) or MMSE, are known for very low complexity, but at the expense of performance loss. Low complexity receiver suitable for STSK are proposed in [5, 6] and the similar approach adjusted to LPSM is used in this paper. Let the outputs of the SM or LPSM coders for all transmit antennas are: u t = f t (B SM )s (31) where t = 1,,, N T is the transmit antenna index, i = 1,,, M is the index of the QAM/PSK symbol inside the SC- FDMA LPSM symbol, B SM =,, b SM,T ) are the vectors of length T of SM bits, f t (B SM ) are the functions for

6 LPSM coding, according to Table II for two transmit antenna case or Table IV for four transmit antenna case, and s are QAM/PSK symbols. For conventional SM, the coding is performed in the same manner with different functions f t (B SM ). Hence, the N T output vectors in the TD can be presented as u t = (u (1) t, u () t,, u (M) t ) T. After the M-point FFT block, U t, that are N T vectors of size M 1 in the FD, are transmitted over N T antennas (after the zero padding, IFFT and CP addition, according to Fig. 1.). It is assumed that the receiver can estimate all N T N R channels, i.e. all N T N R M subchannels, so the receiver can create an N R N T channel matrix H for every subcarrier. If the quasi-static channels are assumed, the received signals on i th subcarrier can be presented as: U 1 U Y = H + N (3) ( U NT) where U t are the i th elements of the vectors U t and N is an N R 1 complex Gaussian white noise vector in the FD for i th subcarrier. The receiver can perform LE by generating matrix W as: W [MMSE] W [ZF] = (H H H ) 1 H H (33) = (H H H SNR I N T ) H H (34) where (33) presents ZF and (34) MMSE equalization and SNR denotes signal-to-noise power ratio, assumed to be estimated by the receiver. For the LE (ZF or MMSE), the receiver performs the equalizations for all M subcarriers independently (where i = 1,,, M is the subcarrier index): U 1 U U = = W Y (35) ( U NT ) and from the obtained results creates vectors U t, that denote the equalized received signals transmitted from the antenna t: U t U t = (36) (M) ( U t ) After the M-point IFFT, (1) u t () u t (1) () U t u t = = F 1 M U t (37) (M) ( u t ) the receiver creates vectors u as: u 1 u u = (38) ( u NT ) As this presents the vector of the equalized received signals from all N T transmit antennas for the i th QAM/PSK symbol inside the SC-FDMA LPSM symbol, the receiver performs the detection using MLD approach: argmin u u u F (39) A LPSM where A LPSM is the set of all possible outputs, N T 1 vectors, of the LPSM coder. If the underlying modulation is of size L and the transmitter has N T = T transmit antennas, the set size is L+T vectors. Obviously, MLD decision in this receiver is performed per LPSM information pair (SM bits and QAM/PSK symbol), not on the whole SC-FDMA LPSM symbol, hence its complexity is low and the set of possible decisions is much smaller compared to the optimal MLD receiver. Despite their simple implementation and low complexity, the receivers with linear equalization are known to have performance loss, compared to the optimal receiver, therefore alternative solutions have to be considered. C. The proposed near-mld receiver As element-wise multiplication in the TD can be written in the FD as a product of a cyclic convolution matrix with a vector, (31) in the FD can be written as: U t = 1 M C ts (40) where C t are M M convolution matrices generated from the M samples in the FD, obtained as F M [f t (B (1) SM ) f t (B () SM ) f t (B (M) SM )] T and S is an M 1 vector obtained from the block of QAM/PSK symbols as S = F M [s (1) s () s (M) ] T. As the channels are assumed to be quasi-static, the received signals, M 1 vectors, can be presented as: N T Y r = 1 M ( diag(h t,r)c t ) S + N r (41) t=1 where r and t are the indices of the receive and transmit antenna, respectively, H t,r is the channel frequency response vector, size M 1, of the channel between t th transmit and r th receive antenna and N r is a complex Gaussian white noise vector. Using vertical vector/matrix concatenation for all receive antennas, from (41): N T Y = 1 M ( H tc t ) S + N (4) t=1 is obtained, where Y is an MN R 1 vector, H t are the channels matrices of size MN R M and N is an MN R 1 noise vector. Using N T H 0 = 1 M H tc t t=1 (43) (4) can be written as: Y = H 0 S + N (44) Hence, LE can be performed by generating the matrices: W ZF = (H H 0 H 0 ) 1 H H 0 (45) W MMSE = (H H 0 H SNR I H M) H 0 (46) Further, from

7 s = F 1 M WY (47) all QAM symbols can be independently detected. It can be noted that in (40) SM bits and QAM symbols are separated, hence in (44) the matrix H 0 depends on SM bits (thus, it is not known at the receiver side) and the vector S on QAM symbols. As H 0 depends on SM bits, the receiver cannot perform this equalization. Fig. 3. presents the block scheme of the proposed receiver. After the LE solution is obtained, the receiver can expect that on M vectors of SM bits vectors B SM, i = 1,,, M, the possible number of errors on SM bits vectors is less than or equal to e. In other words, up to e, out of M, SM bits vectors are assumed to have an error. As B SM carries T bits, it can have N T possible values. Hence, from the detected SM bits e M! sequence, z = m=1 ( (N m!(m m)! T 1) m ) different sequences can be generated. E.g. for e = 1, z = M(N T 1) sequences can be generated. Further, in the case of N T =, two transmit antenna case, z = M sequences with one different SM bit, compared to the LE solution, can be generated. (improved MMSE) receiver. The main reason for the proposed near-mld receiver increased complexity, as it will be discussed in Section IV, is z times performed LE. To avoid this, for immse receiver, we propose using only one LE and creating the reduced set of decisions using the equalized signals obtained after this LE. Fig. 4. presents the block scheme of the proposed immse receiver. After obtaining an LE solution, using (39), the receiver can make a similar assumption as for the previous receiver, that a number of the errors in SM bits (two transmit antenna case) or in the vectors of SM bits (four transmit antenna case) is less than or equal to e. From (38), u = (u 1 u u NT ) T presents the equalized signals on i th position in the TD from all transmit antennas. Let u = (u 1 u u NT ) T present the MLD decision, obtained from (39), as one of L+T possible transmit vectors that is the nearest to the u. Further, u provides the information of QAM/PSK symbol s and SM bits in B SM, for each i. Fig. 4. The block scheme of the proposed immse receiver. Fig. 3. The block scheme of the proposed near-mld receiver. For the each of z sequences, the receiver performs (43) to (47). First, it creates channel matrix H 0 (43) because matrices C t depend on SM bits vectors only. Further, it creates equalization matrices W (using (45) or (46)), then performs z times LE. After all z solutions are obtained, the receiver selects the decision with the minimum distance (48): Y H (j) 0 F M s (j) (48) F where j = 0,1,, z is the index of the sequence (j = 0 is the LE solution and other z are the sequences created from the LE solution). The selected sequence provides SM bits, whereas QAM decisions are obtained from the corresponding LE. This approach reduced the set of decisions from M(L+T) of MLD to z + 1. The set size depends on the parameter e. For large e, the size of the set of decisions is increased, resulting in increased receiver complexity, but the performance approach close to MLD. For low e, the complexity is substantially decreased (e.g. e = 1 and N T = implies z = M, so the set size grows linearly with M). The size of the set of decisions does not depend on the constellation size. Although the set of decisions is reduced, the receiver has to perform z + 1 LE equalizations. z times performed matrix multiplication and inversion, in (45) or (46), of an MN R M matrix is still complex for a large number of the subcarriers. However, it can be used for a small number of the subcarriers. D. Proposed immse receiver Due to significantly increased complexity of the proposed near-mld receiver when the number of subcarriers is increased, we propose another receiver, denoted as immse Using the assumption of the errors on SM bits, the receiver can make additional N T 1 decisions for QAM/PSK symbols from u, by observing all other SM bits possibilities. Let B SM,f and s f present the additional N T 1 decisions, where. f = 1,, N T 1, for all possible SM bits vectors except B SM After obtaining these solutions, the receiver creates z = M! e m=1 ( (N m!(m m)! T 1) m ) sequences from the obtained LE solution sequence by placing up to e additional decisions (B SM,f and s f ) on the corresponding positions instead of LE obtained decisions. After that, the LE solution sequence and z created sequences are LPSM coded, according to Table II or Table IV. After an M-point FFT, z + 1 vectors of length M in the frequency domain are obtained for each transmit antenna. Hence, let U t,j, for j = 0,1,, z, present z + 1 vectors of length M in the FD for all N T transmit antennas. After that, the receiver creates vectors U t,j of length N T as: U t,j = U 1,j U,j ( U NT,j ) (49) where U t,j are i th elements of the vectors U t,j. If Y presents an N R 1 vector of the N R received signals in the FD on the i th subcarrier, N R N T channel matrix H for the observed i th subcarrier, then, using the MLD approach, the one that minimizes the distance:

8 argmin j=0,1,,z M Y H U t,j F i=1 (50) is selected as the final decision. The obtained index j is used for the selection of the one of z + 1 QAM/PSK sequences and corresponding SM sequences. Unlike the proposed near-mld receiver, this receiver performs LE only once. However, as with near-mld receiver, the set of decisions grows for greater e. Therefore, in the simulations, it is assumed that e = 1, hence z = M(N T 1). This implies that after the LE solution is obtained, the receiver creates additional z = M(N T 1) sequences. C nearmld ZF = C ZF + (N R + N T N R + 1)zM 3 + (N T + N R + )zm + N R M (53) C nearmld ZF = (N R + N T N R + 1)zM 3 + (N T z + zn R + z + N T )M + (N 3 T + N (54) T N R + N R )M + N T (L+T) or, when MMSE is used: C nearmld MMSE = C MMSE + (N R + N T N R + 1)zM 3 + (N T + N R + )zm (55) + (N R + z)m IV. LSPM RECEIVER COMPLEXITY In order to evaluate the complexity of the receivers, the number of multiplications is compared. Here, the multiplication is observed as a multiplication of two complex scalars. Operations, such as addition, reordering, transpose, Hermitian, conjugate etc. are not observed. Also, the common part for all receivers, such as cyclic prefix removal, N-point FFT or channel estimation (and SNR estimation for MMSE receiver), is not observed too. We assumed that the matrix inverse for n n matrix takes n 3, M-point FFT or IFFT takes M and Frobenious norm for n m matrix takes nm multiplications. LPSM coding, according to the Tables II and IV, and QAM/PSK modulation are not considered as multiplications. A. MLD receiver First of all, optimal MLD receiver is observed. As the set of the possible transmitted signals in the FD depends on the number of subcarriers, M, the receiver has to calculate the distance from the received signals for all possible transmitted signals. Hence, the complexity is: C MLD = MN R (N T + 1) M(L+T) (50) Obviously, the main problem is that the complexity grows exponentially with M. B. LE receiver LE receiver has to compute (3) or (33) for all M subcarriers. The matrices H are of sizes N R N T, hence for H H H the receiver performs N T N R multiplications. Therefore, the number of the multiplications for ZF receiver per SC-FDMA LPSM symbol is: C ZF = N T M + M(N 3 T + N T N R ) + N T (L+T) (51) And, analogously for MMSE: C MMSE = N T M + M(N 3 T + N T N R + N T ) (5) + N T (L+T) Hence, it can be noted that the LE receiver complexity is O(M ). This part exists because of M-point IFFT performed for all N T antennas after the LE. C. Near-MLD receiver After performing LE, this receiver creates additional z sequences for SM bits vectors and performs z times LE. When all z + 1 solutions are obtained, the receiver performs MLD to select the one with the minimum distance from the received signal. It can be shown that its complexity, when ZF is used, is: C nearmld MMSE = (N R + N T N R + 1)zM 3 + (N T z + zn R + z + N T )M + (N R + z + N 3 T + N (56) T N R + N T )M + N T (L+T) For this receiver, the complexity depends on the parameter z that depends on M. As z = m=1 ( (N m!(m m)! T 1) m ), it can be noted that for greater e the complexity grows significantly. Because of that, in the simulations, we assumed that e = 1. Hence, z = M(N T 1) and (54) and (56) can be written as C nearmld ZF = (N R + N T N R + 1)(N T 1)M 4 + (N T + N R + )(N T 1)M 3 + N T M (57) + (N 3 T + N T N R + N R )M + N T (L+T) C nearmld MMSE = (N R + N T N R + 1)(N T 1)M 4 + (N T + N R + )(N T 1)M 3 + (N T 1)M (58) + (N R +N 3 T + N T N R + N T )M + N T (L+T) Obviously, the difference between MMSE and ZF, from the complexity point of view, is very small. In both cases the complexity is O(M 4 ). D. immse receiver After performing LE, this receiver creates additional z sequences for SM bits vectors and performs z additional QAM/PSK detections. This avoids z additional LE performed for near-mld receiver. It can be shown that its complexity is: C immse = C MMSE + zn T M + N T L + N T M 3 (59) + (z + 1)N R N T M Using (51) and the assumption that e = 1, (59) can be written: C immse = N T M 3 + N T M + (N 3 T + N T N R + N T + N R N T )M (60) + N T L ( T + 1) Therefore, the complexity is O(M 3 ), which is smaller than near-mld receiver, but greater than simple MMSE receiver. V. COMPARISON AND SIMULATION OUTCOMES A. PAPR comparison First of all, PAPR for SC-FDMA with SM and LPSM are compared with the conventional SC-FDMA signal. Discrete- e M!

9 time PAPR [8, 9] can be defined as a ratio of the peak and average power on a block of samples in the TD. If the discrete-time sampled transmitted signal is presented as a vector x = [x 1, x,, x N ] T, then PAPR of the signal can be defined as: max x n 1 n N PAPR(x) = (61) E{ x n } The PAPR is usually measured using the CCDF (Complementary Cumulative Distribution Function) [8]. CCDF expresses the probability that the PAPR is greater than the level PAPR 0 : CCDF(PAPR 0 ) = P(PAPR > PAPR 0 ) (6) For the comparison, QPSK underlying modulation is used and the user occupied M = 60 subcarriers (or 5 LTE resource blocks). four transmit antenna case, SM shows even higher PAPR than OFDM. In addition, the slight PAPR decrease in LPSM over conventional SC-FDMA can be observed in Fig. 6. The reason for this is that due to the phase shift α the original underlying constellation QPSK appears as 16-PSK constellation, resulting in the observable PAPR reduction. B. Performance comparison For the performance comparison, different cases have been observed. In all cases, perfect channel estimation and synchronization are assumed. Power back-off and forward error correction are not used. In the first comparison, two transmit antenna case is observed and we compared all mentioned LPSM receivers: MLD, near-mld, immse, MMSE and ZF. In addition, Alamouti-based [30] STBC implemented in SC-FDMA is included for the performance comparison with another wellknown MIMO technique. STBC is chosen as it has similar transmitter complexity (the same number of RF chains), but achieves transmit diversity rather than bitrate increase. As MLD receiver has a huge computational complexity, for this comparison, the case with only M = 4 subcarriers is observed. MIMO x4 with uncorrelated channels modeled as EVA-5 [31] is observed. LPSM cases use QPSK as the underlying modulation, and because N T =, it results in 3 bits transmitted per subcarrier in one SC-FDMA symbol. For fair comparison, STBC uses 8PSK as the underlying modulations. LTE 5 MHz channel is used, hence N = 51. For the proposed receivers, near-mld and immse, e = 1 is used to decrease the set of decisions and receiver complexity. Fig. 5. The PAPR comparison for SC-FDMA SM and LPSM in the case of two transmit antennas with SC-FDMA and OFDM. The user occupies 5 LTE resource blocks (60 subcarriers) Fig. 7. The performance comparison for SC-FDMA LPSM with different receivers and STBC in x4 MIMO uncorrelated EVA-5 channels. M = 4. SC- FDMA symbol carries 3 bits per subcarrier. Fig. 6. The PAPR comparison for SC-FDMA SM and LPSM in the case of four transmit antennas with SC-FDMA and OFDM. The user occupies 5 LTE resource blocks (60 subcarriers) Fig. 5. presents PAPR curves for two transmit antenna case, and Fig. 6. for four transmit antenna case. Obviously, LPSM maintains low PAPR level on all antennas and conventional SM shows a significant increase. For two transmit antenna case, SM PAPR is still better than OFDM PAPR, whereas for As can be seen in Fig. 7., LPSM with MLD, which is the optimal receiver, and LPSM with near-mld receiver shows the best performance. Near-MLD receiver shows performance the same as MLD (their two lines overlap) and, as it is shown in Section IV, with significantly lower complexity. MMSE and ZF LPSM receivers are of significantly deteriorated performance, compared to MLD, but these receivers have very low complexity. immse presents a tradeoff between the good performance and low complexity. In general, when compared

10 to STBC, LPSM shows very good performance, even with low complexity receivers. For the second performance comparison, very complex MLD receiver is excluded and more realistic scenario with M = 36 subcarriers (or 3 LTE resource blocks) is observed. Near-MLD, immse, MMSE and ZF receivers for LPSM and Alamouti-based [30] space-frequency block coding (SFBC) implemented in SC-FDMA, are compared. In addition, although it has been confirmed that conventional SM increases PAPR, the comparison is also given against SM MMSE and ZF receivers. These receivers are exactly the same as LPSM MMSE and ZF with the difference in the coding functions f t (B SM ). The channel model is ETU-70 [31], with more severe conditions, and all other parameters and assumptions are the same as in the previous comparison. As it can be seen in Fig. 8., LPSM with the proposed receivers, near-mld or immse, again outperforms SFBC, a transmit diversity technique, for the same bitrate. In addition, simple MMSE is even better than SFBC, whereas ZF is very near to SFBC. It is worth noting that LPSM with simple LE receivers shows better performance than SM with the same receivers. Fig. 8. The performance comparison for SC-FDMA LPSM with different receivers and SFBC in x4 MIMO uncorrelated ETU-70 channels. M = 36. SC-FDMA symbol carries 3 bits per subcarrier. Fig. 9. The performance comparison for SC-FDMA LPSM with different receivers and QO-SFBC in 4x8 MIMO uncorrelated EPA-5 channels. M = 96. SC-FDMA symbol carries 4 bits per subcarrier. STBC, SFBC and QO-SFBC are provided only for the performance comparison and their PAPR was not analyzed. The issue of low PAPR in transmit diversity techniques in SC- FDMA has been studied [33-35]. C. Complexity comparison Using the results from Section IV, the complexity comparison of the receivers is presented in Fig. 10. as the function of the number of subcarriers, M. For the comparison, it has been assumed that the transmitter has four transmit antennas (N T = 4), the receiver has eight receive antennas (N R = 8) and the underlying modulation is QPSK (L = ). As mentioned before, e = 1, hence z = M(N T 1). It can be noted that near-mld receiver, which provided the performance very near to MLD receiver, as it can be seen in Fig. 7, has much lower complexity than MLD, but still requires large number of multiplications, as it grows as O(M 4 ). Linear receivers, MMSE and ZF, have very low complexity and immse presents a tradeoff between linear and optimal receivers. As it is shown in Section IV, near-mld receiver has significantly greater complexity than immse, hence its complexity can present a problem for the greater number of the subcarriers. In the last performance comparison, the case with M = 96 subcarriers (or 8 LTE resource block) and four transmit antenna (N T = 4) is observed. For LPSM, QPSK underlying modulation is used and, in order to have a fair comparison, quasi orthogonal SFBC (QO-SFBC) [3] with 16-QAM is observed. The channel is modeled as 4x8 MIMO with uncorrelated EPA-5 [31] channels. The outcomes, presented in Fig. 9., show that LPSM outperforms QO-SFBC even with simple LE receivers and that immse offers significant improvement in comparison to MMSE or ZF. Fig. 10. The complexity comparison for different SC-FDMA LPSM receivers for 4x8 MIMO, QPSK modulation and e = 1.

11 VI. CONCLUSION In this paper, we presented LPSM, an SM alternative that preserves low PAPR of SC-FDMA. The coding scheme is obtained for two and four transmit antenna cases. Due to inherently high complexity of optimal MLD receiver in SC- FDMA, linear receivers are observed and two new receivers are proposed near-mld and immse receiver. They are based on a fact that SM bits and QAM/PSK can be separately obtained. Therefore, the number of the decisions used for MLD can be significantly decreased. It has been shown that near-mld receiver achieves very similar performance as MLD with significantly decreased complexity. immse presents a tradeoff between linear receivers (ZF and MMSE) and near-mld. At the expense of performance, the complexity is significantly decreased, when compared to near-mld receiver. 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